Canonical matrices of isometric operators on indefinite inner product spaces

Abstract

We give canonical matrices of a pair (A,B) consisting of a nondegenerate form B and a linear operator A satisfying B(Ax,Ay)=B(x,y) on a vector space over F in the following cases: (i) F is an algebraically closed field of characteristic different from 2 or a real closed field, and B is symmetric or skew-symmetric; (ii) F is an algebraically closed field or the skew field of quaternions over a real closed field, and B is Hermitian or skew-Hermitian with respect to any nonidentity involution on F. We use a method that admits to reduce the problem of classifying an arbitrary system of forms and linear mappings to the problem of classifying representations of some quiver. This method was described in [V.V. Sergeichuk, Math. USSR-Izv. 31 (1988) 481-501].

57 pages

References in corpus (7)

• Canonical matrices for linear matrix problems
• Canonical forms for complex matrix congruence and *congruence
• Classification problems for system of forms and linear mappings
• Complexity of matrix problems
• Congruences of a square matrix and its transpose
• A regularization algorithm for matrices of bilinear and sesquilinear forms
• Unitary and Euclidean representations of a quiver

Cited by in corpus (8)

• Canonical matrices of bilinear and sesquilinear forms
• Congruence of matrix spaces, matrix tuples, and multilinear maps
• Classification of linear operators satisfying $(Au,v)=(u,A^{r}v)$ or $(Au,A^{r}v)=(u,v)$ on a vector space with indefinite scalar product
• Isometric and selfadjoint operators on a vector space with nondegenerate diagonalizable form
• Symplectic spaces and pairs of symmetric and nonsingular skew-symmetric matrices under congruence
• Classification of linear mappings between indefinite inner product spaces
• Lipschitz property for systems of linear mappings and bilinear forms
• Canonical matrices of forms and pairs of forms over finite and p-adic fields